# Homework Help: Calculating the residue of a complex function

1. Apr 12, 2012

### AlBell

1. The problem statement, all variables and given/known data

calculate the residue of the pole at z=i of the function

f(z)=(1+z^2)^-3

State the order of the pole

2. Relevant equations

I know the residue theorem and also the laurent series expansion but I'm having trouble applying these

3. The attempt at a solution

I think the pole is order 3 but I'm having trouble either expanding the function in order to find the a(-1) term or implementing the residue theorem and would appreciate some help! Thanks

2. Apr 12, 2012

### micromass

Do you know that the residue of a function f at the point a can be calculated by

$$\lim_{z\rightarrow a} (z-a)f(z)$$

3. Apr 12, 2012

### AlBell

I thought it was ?

4. Apr 12, 2012

### micromass

Yes, of course, sorry. My formula is for a simple pole.
So, can you use your formula to find the residue?

5. Apr 12, 2012

### AlBell

Yes but I am having trouble actually implementing the formula and wondered if anyone could do an example to help me understand it?

6. Apr 12, 2012

### micromass

Well, first you'll need to determine the order of the pole.

Your guess is that the order of the pole is 3. So you'll need to look at

$\lim_{z\rightarrow a} (z-a)^3f(z)$.

This limit should be nonzero for the order of the pole to be 3. If it is zero, then the pole is of a lower order. If the limit doesn't exist, then the pole is of higher order.

So, can you calculate that limit?

Hint: factor the denominator.

7. Apr 13, 2012

### AlBell

Ah I forgot about factorising, that makes it so much simpler! I get 3/8 I think!

8. Apr 13, 2012

### Robert1986

VERY close. You probably got something like 12*(2i)^(-5), right? Just compute this again. The 3/8 is right, but you are missing two factors.